We show that the Simpson’s rule integrates exactly polynomials of degree ≤ 3 then from equation of error we show that
R2(f, x) = 0 for f(x) = 1, x, x2, x3.
Substituting f(x) = 1, x, x2, x3 in equation
we get,
Hence, the Simpson’s rule integrates exactly polynomials of degree ≤ 3. Therefore, the method is of order 3. It is interesting to note that the method is one order higher than expected, since we have approximated f(x) by a polynomial of degree 2 only
Let f(x) = x4.
from equation,
we get,